We carry out the numerical simulations to evaluate Theorem REF .
Following [1]}, the parameter values take a linear form.
Specifically, we set \(\alpha _{i+1}^{*}=(m-1-i)L/(m-1)\)  for \(i=0,1,\dots ,m-1\) ,
and considered four different values 0, \(\log (\log (m))\) , \(\log (m)^{1/2}\)  and \(\log (m)\)  for \(L\) .
For the parameter vector \(\beta \) , let \(\beta _{j+1}^{*}=(n-1-i)L/(n-1), j=0,1,\dots ,n-2\)  for simplicity and \(\beta _{n}^{*}=0\)  by default.
